Pricing exotic options in a path integral approach

Abstract

In the framework of the Black-Scholes-Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price European path-dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the cases of Asian, barrier knock out, reverse cliquet and basket call options, evaluating prices and Greeks. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at-the-money and out-of-the-money options, the path integral approach exhibits competitive performances.

Path integral pricing of Wasabi option in the Black–Scholes model

“…That the solution to the path integral (8) is also the fundamental solution to (9) is basically the content of the celebrated Feynman-Kac theorem [15]. Define the propagator W P (X T , T, λ|X 0 , 0) for a driftless risky asset X t that spends λ unit of time over the constant level b…”

Multiplicative noise, fast convolution and pricing

“…With the adoption of this approach, the computational time is noticeably reduced: each FFT computation requires O(m × log 2 (2m)) operations. Finally, in order to compute equation (12) we need to repeat the algorithm at each time step, and on the whole the computational burden can be estimated to be of order O(n × m × log 2 m), which is definitely a satisfactory improvement with respect to the non-FFT based procedure.…”

Pricing exotic options with L-stable Padé schemes

“…Exotic options are widely used in the field of finance (see Taleb, 1997;Joshi, 2003;Bormetti et al, 2006 andLasserre et al, 2006). Exotic options are particularly challenging 0378-4266/$ -see front matter Ó 2007 Elsevier B.V. All rights reserved.…”